A geometric Brownian motion (GBM) (occasionally, exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion[1], also called a Wiener process. It is applicable to mathematical modelling of some phenomena in financial markets. It is used particularly in the field of option pricing because a quantity that follows a GBM may take any positive value, and only the fractional changes of the random variate are significant. This is a reasonable approximation of stock price dynamics except for rare events.
A stochastic process St is said to follow a GBM if it satisfies the following stochastic differential equation:
where Wt is a Wiener process or Brownian motion and μ ('the percentage drift') and σ ('the percentage volatility') are constants.
For an arbitrary initial value S0 the equation has the analytic solution
which is a log-normally distributed random variable with expected value 
and variance 
The correctness of the solution can be verified using Itō's lemma. The random variable log(St/S0) is normally distributed with mean (μ − σ2 / 2)t and variance σ2t, which reflects the fact that increments of a GBM are normal relative to the current price, which is why the process has the name 'geometric'.
References
- ^ Introduction to Probability Models by Sheldon M. Ross, 2007 Section 10.3.2
See also
External links
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